Electronic Journal of Linear Algebra
Volume 16
Article 27
2007
A relation between the multiplicity of the second
eigenvalue of a graph Laplacian, Courant's nodal
line theorem and the substantial dimension of tight
polyhedral surfaces
Tsvi Tlusty
[email protected]
Follow this and additional works at: http://repository.uwyo.edu/ela
Recommended Citation
Tlusty, Tsvi. (2007), "A relation between the multiplicity of the second eigenvalue of a graph Laplacian, Courant's nodal line theorem
and the substantial dimension of tight polyhedral surfaces", Electronic Journal of Linear Algebra, Volume 16.
DOI: http://dx.doi.org/10.13001/1081-3810.1204
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Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 16, pp. 315-324, October 2007
ELA
http://math.technion.ac.il/iic/ela
A RELATION BETWEEN THE MULTIPLICITY OF THE SECOND
EIGENVALUE OF A GRAPH LAPLACIAN, COURANT’S NODAL
LINE THEOREM AND THE SUBSTANTIAL DIMENSION OF
TIGHT POLYHEDRAL SURFACES∗
TSVI TLUSTY†
Abstract. A relation between the multiplicity m of the second eigenvalue λ2 of a Laplacian on
a graph G, tight mappings of G and a discrete analogue of Courant’s nodal line theorem is discussed.
For a certain class of graphs, it is shown that the m-dimensional eigenspace of λ2 is tight and thus
defines a tight mapping of G into an m-dimensional Euclidean space. The tightness of the mapping
is shown to set Colin de Verdiére’s upper bound on the maximal λ2 -multiplicity, m ≤ chr(γ(G)) − 1,
where chr(γ(G)) is the chromatic number and γ(G) is the genus of G.
Key words. Graph Laplacian, Tight Embedding, Nodal domains, Eigenfunctions, Polyhedral
Manifolds.
AMS subject classifications. 15A18, 05C10, 47A75, 15A90.
1. Introduction. Laplacians, discrete or continuous, are omnipresent in physics
and mathematics and much effort has been directed to analysis of their spectra [8, 10,
16, 14, 13]. A seminal result in this field is Courant’s nodal line theorem that relates
the order of the Laplacian’s eigenvalues to the sign patterns of the corresponding
eigenfunctions [15, 27]: If the eigenfunctions of a Laplacian on a domain are ordered
according to increasing eigenvalues, then the nodes of the n-th eigenfunction divide
the domain into no more than n nodal domains. The nodal domains, in which the
eigenfunction takes one sign, are separated by the nodal sets that are the zero level-sets
of the eigenfunction. For graph Laplacians, the discrete analogue of the nodal domain
becomes the sign-graph, a maximal connected subgraph on which an eigenfunction
takes the same sign. On weak sign-graphs the eigenfunction is either non-positive
or non-negative, while on strong sign-graphs the sign of the eigenfunction is strictly
positive or negative. Several authors have found the following discrete analogue of
Courant’s nodal line theorem [13, 17, 21, 20]:
Theorem 1.1. On a connected graph G, the n-th eigenfunction un of the Laplacian ∆ has at most n weak sign-graphs.
For the special case of the second eigenvalue λ2 , it was shown earlier that the
corresponding eigenfunction u2 cuts the graph into exactly two weak sign-graphs
[18, 19]. Among the eigenvalues of the graph Laplacian, λ2 plays a special role.
Physically, it corresponds to the mixing time of a Brownian walk on the graph and to
its first-excited energy level. Mathematically, λ2 was shown to be related to several
structural and isoperimetric properties of the graph, such as the max-cut problem
[2, 1, 28].
∗ Received by the editors 24 March 2007. Accepted for publication 30 September 2007. Handling
Editor: Richard A. Brualdi.
† Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100,
Israel ([email protected]).
315
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Tsvi Tlusty
Of special interest is m, the multiplicity of λ2 and the dimension of the λ2 eigenspace. m is the degeneracy of the first-excited energy level, which is the first
mode to appear at several types of continuous (or second-order) phase transitions.
Continuous phase transitions are known to occur in noisy information channels when
the signal distortion level is varied [6] and have been related to certain classification
and optimization problems [30]. We suggested that this type of phase transition
may occur during the Darwinian evolution of noisy biological information channels,
and may be the mechanism underlying the emergence of codes in these channels
[33, 34, 23]. Noisy information channels may be described in terms of error-graphs,
in which edges connect two signals that are likely to be confused by noise. The
Laplacian of the error-graph is the operator that measures the average effect of errors
and therefore controls the phase transition. This motivated the present letter that
focuses on the relation between the topology of the error-graph and the maximal
degeneracy of the Laplacian’s first-excited modes.
Colin de Verdiére revealed an intimate relation between the topology of a Riemannian surface S and the supremum of m over all possible Laplacians on the surface
m̄(S) through the chromatic number chr(S) [14, 13, 11, 12]:
Theorem 1.2. For any surface S, the maximal multiplicity of the Laplacian’s
second eigenvalue is bounded from below, m̄(S) ≥ chr(S) − 1.
The chromatic number of a surface chr(S) is defined as the least number of colors
required to color any map on S. The dependence of the chromatic
on the
number
√
topology of S is given by Heawood’s formula [29], chr(γ) = 12 7 + 49 − 24χ ,
where χ(S) is the Euler characteristic of the surface (the genus of an oriented surface
is γ(S) = 1−χ(S)/2). Colin de Verdière conjectured that the lower bound of Theorem
1.2 is exactly the maximal possible multiplicity, m̄(S) = chr(S) − 1 [12].
The multiplicity m̄(S) was shown to be bounded also from above [9]. In a series
of improvements [7, 26, 31], this upper bound approached the lower bound, m̄(S) ≤
5 − χ(S). The last improvement is due to Séveneec, who also brought up a potential
link between the multiplicity m̄(S) of Laplacians on a surface S and tight mappings
of S into Euclidean spaces. These mappings are termed tight because no hyperplane
can cut the mapped surface into more than two pieces, which gave tightness its other
name, the two-piece property. Banchoff showed that if S is a polyhedral surface, then
the substantial dimension of such tight mappings d is bounded , d ≤ chr(S) − 1 [3].
The equality of Banchoff’s and Colin de Verdiére’s bounds is not a mere coincidence. The purpose of the present letter is to elucidate this relation between tight
mappings, the maximal multiplicity of the second eigenvalue of graph Laplacians and
Courant’s nodal line theorem, and to use this relation to prove that Colin de Verdière’s
bound applies for a rather wide class of graphs (Theorem 3.1).
In Section 2 we list the basic notation and discuss the relation between tightness
and the nodal pattern of the λ2 -eigenfunctions. We show that the λ2 -eigenspace of
certain graphs, namely paths, cycles, complete graphs and their Cartesian products
are tight. In Section 3 we prove Colin de Verdière’s bound for the class of graphs
whose λ2 -eigenspaces are tight. The sketch of the simple proof is the following: We
consider a graph G that is embedded into a polyhedral surface S. The λ2 -eigenspace
of the graph Laplacian is m-dimensional and defines a tight mapping of G and S
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The Multiplicity of the Second Eigenvalue of a Graph Laplacian
317
into the m-dimensional Euclidean space. This sets Banchoff’s upper bound on the
dimension of the Euclidean space, which by the construction of the mapping is also
the maximal multiplicity, m ≤ chr(S) − 1 = chr(γ(G)) − 1.
2. Tightness and the λ2 -eigenfunctions of a Laplacian. Before we discuss
the question of which λ2 -eigenspaces are tight, we list some basic notations related to
the graph Laplacian and to the mapping induced by its eigenfunctions. We consider
an undirected, simple, loop-free graph G with a vertex set V of N vertices and an
edge set E. The vertices are denoted by i = 1, 2 . . . N . The graph Laplacian ∆ is an
operator which may be expressed as a real symmetric matrix ∆ ∈ RN ×N associated
with the graph in the following way: If vertices i and j are adjacent, (i, j) ∈ E,
then the corresponding entry in the Laplacian is negative, ∆ij = ∆ji < 0, otherwise
and the diagonal terms assure that the sum over rows and columns vanishes,
∆ij = 0, ∆ii = − j=i ∆ij . The operator ∆ is sometimes termed a weighted Laplacian while
’Laplacian’ is reserved for the case where the negative entries are all ∆ij = −1. ∆ is
irreducible iff the associated graph is connected. By the Perron-Frobenius theorem,
the first eigenvalue λ1 = 0 is non-degenerate and the λ1 -eigenfunction may be chosen
to be the all-ones vector 1. The multiplicity of the second eigenvalue λ2 is denoted by
m. Similar to the continuous case, one may define the maximal λ2 -multiplicity over
all possible Laplacians on a graph, which is denoted by m̄(G).
Every graph G can be embedded in a surface S by a continuous one-to-one mapping (homeomorphism) i : G → S. Intuitively, this means that G can be drawn on
S with no intersecting edges. The genus of a graph γ(G) is the minimal genus of a
surface S into which G can be embedded ([22], chapter 3). The surface S can in turn
be embedded into a Euclidean space Rd by another homeomorphism ϕ : S → Rd . A
polyhedral surface is a surface embedded in Rd such that its image ϕ(S) is a finite
union of planar polygons, termed faces. If ϕ is only locally one-to-one it defines a
polyhedral immersion, in which the polygonal faces can intersect. An embedded or
immersed surface S is substantial if it is not contained in any hyperplane.
An em
bedded surface S is tight if its intersection with any half-space S h is connected.
Similarly, animmersion S is tight if the preimage of the intersection with any halfspace ϕ−1 (S h) is connected ([24], chapter 2). Functions (or vectors) on a graph are
also maps, from the vertices into the real numbers, u : V → R and we can therefore
apply a similar notion of tightness:
Definition 2.1. Let u be a function on a graph G and s ∈ R a level. Let G+ (u, s)
and G− (u, s) the subgraphs induced by the vertex sets V+ (u, s) = {i ∈ V |u(i) ≥ s}
and V− (u, s) = {i ∈ V |u(i) ≤ s}, respectively. If for any level s ∈ R both G+ (u, s)
and G− (u, s) are connected (or empty), then the function u is tight. A vector space
of functions is tight if it contains only tight functions.
The intuitive link between tightness and λ2 -eigenfunctions is evident. The λ2 eigenfunctions cut graphs into two weak sign-graphs, one non-negative and one nonpositive, while tight surfaces are cut by hyperplanes into no more than two pieces.
Trying to apply this intuition, one notes an obstacle: Nodal domains are determined
by zero level-sets, that is by hyperplanes that pass through the origin 0, while tightness
is a somewhat stricter condition on every affine hyperplane, not only those that pass
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1
0.5
0
−0.5
−1
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20
1
0.5
0
−0.5
−1
Fig. 2.1. Two functions on the cycle graph C20 . The x coordinate specifies the vertex and the
y coordinate is the value of the function at this vertex. Both functions are cut into two sign-graphs
by the zero-level set (s = 0, solid line) and could be λ2 -eigenfunctions by Theorem 1.1. However,
the lower function is not tight, because it is cut by several level-sets (e.g. s = ± 21 , dashed lines)
into more than two pieces.
through the origin. This difference is demonstrated in Figure 2.1, where two functions
on the cycle graph C20 are plotted. Both functions are cut into two sign-graphs by
the zero-level set, and can therefore be, in principle, λ2 -eigenfunctions of a Laplacian
on C20 . However, one of the functions is not tight, because it can be cut by certain
nonzero level-sets into more than two pieces. In this simple example, tightness was
lost by the presence of more than two extrema.
An immediate corollary of Definition 2.1 is:
Corollary 2.2. Any function u on a complete graph Kn is tight.
This is because all the vertices in a complete graph are connected to each other.
It is evident from Definition 2.1 that a tight function can have only one minimum
and one maximum (Figure 2.1). Using matrix reducibility arguments, Fiedler showed
that λ2 -eigenfunctions of graph Laplacians have a property related to tightness but
somewhat weaker ([19], Theorem (3,3)):
Lemma 2.3. If u is a λ2 -eigenfunction of a Laplacian of a connected graph G
then (a) G+ (u, s) is connected for any s ≤ 0 (b) G− (u, s) is connected for any s ≥ 0.
We use Lemma 2.3 to prove that λ2 -eigenspaces of cycles and paths are always
tight.
Lemma 2.4. If the maximal degree of a connected graph G is 2, that is G is a
cycle or a path, then the λ2 -eigenspace of a Laplacian on G is tight.
Proof. Assume that a λ2 -eigenfunction u is not tight. This means that for some
s one of the induced subgraphs G+ (u, s) and G− (u, s) is not connected. Without loss
of generality, assume that s > 0 and by Lemma 2.3 G− (u, s) is a connected path. If
G is a cycle then G+ (u, s) must also be connected and u is therefore tight (Figure
2.2, left).
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The Multiplicity of the Second Eigenvalue of a Graph Laplacian
319
G+(u,s)
G-(u, s)
G-(u,s)
G+(u,0)
G+(u,s)
G+(u, s)
Fig. 2.2. The partition of a cycle (left) and a path (right) by the subgraphs induced by a level
s > 0 and the λ2 -eigenfunction u (proof of Lemma 2.4). The induced subgraph G− (u, s) is connected
by Lemma 2.3. If Gis a cycle then G+ (u, s)) must also be connected and u is therefore tight (left).
If G is a path, then by Courant’s Theorem 1.1 G+ (u, 0) is connected and therefore must contain
G− (u, s) (right). It follows that the eigenfunction u is tight (see proof ).
If G is a path, then G− (u, s) separates the two components of G+ (u, s) (G+ (u, s)
cannot have more than two components since G− (u, s) has no more than two edges at
its boundary). By the definition of the induced subgraphs (Definition 2.1), G+ (u, s) ⊆
G+ (u, 0) and G− (u, 0) ⊆ G− (u, s). By Theorem 1.1 G+ (u, 0) is connected and therefore must contain G− (u, s) (Figure 2.2 right). It follows that G− (u, 0) ⊆ G− (u, s) ⊆
G+ (u, 0). G− (u, 0) and V− (u, 0) must therefore be empty, or contain only vertices
on
which u = 0. However, since u is orthogonal to the λ1 -eigenfunction, 1, i.e.
v u(v) = 0, it must contain negative entries and G− (u, 0) is nonempty. The eigenfunction u must therefore be tight.
Returning to our previous example in Figure 2.1, it follows from Lemma 2.4 that
the lower function that is not tight cannot be a λ2 -eigenfunction of a Laplacian on the
cycle graph, although it has only two sign-graphs. The presence of a minimum between
the two maxima allows the s = 12 level set to cut G− (u, 12 ) into two components, thus
contradicting Lemma 2.3. Similarly, the s = − 21 level-set cuts the G+ (u, − 12 ) into
two components, again contradicting Lemma 2.3.
The only possible critical points on paths and cycles are maxima and minima.
This is because these graphs are inherently unidirectional and to identify a saddle
point at a vertex one needs at least two directions and more than two neighbors. A
saddle point could separate the two components of G+ (u, s > 0) keeping G− (u, s > 0)
connected (but not simply-connected). Thus, the presence of a saddle point allows a
λ2 -eigenfunction that is not tight. We remark that if the graph is a 1-skeleton of a
polyhedral surface then the numbers of critical points are related by Morse’s formula
[4], |maxima| − |saddles| + |minima| = χ = 2 − 2γ. Therefore, for tight functions
|maxima| = |minima| = 1 and |saddles| = 2γ.
Cartesian graph products appear in various fields, such as coding and information
theory [33, 34, 23, 32]. The Cartesian product P = GH of the two graphs G and H
has the vertex set VP = VG ⊗ VP = {(g, h)|g ∈ VG , h ∈ VH }, that is all the ordered
pairs of vertices from G and H . Two vertices (g1 , h1 ), (g2 , h2 ) are adjacent iff g1 = g2
and h1 , h2 are adjacent in H, or h1 = h2 and g1 , g2 are adjacent in G. The Laplacian of
the graph product is given by ∆P = ∆G ⊗ IH + IG ⊗ ∆H , where ⊗ is the Kronecker
tensor product and IG , IH are the identity matrices of G and H, respectively. It
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1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
Fig. 2.3. The z coordinate represents the λ2 -eigenfunction uP = 1 ⊗ uC20 on the product
P = C10 C20 . The ’projection’ of the eigenfunction on C20 (thick solid line) is uC20 , the λ2 eigenfunction of C20 .
follows that the eigenfunctions of ∆P are also products of the eigenfunctions of ∆G
and ∆H , uP = uG ⊗ uH with the corresponding eigenvalues λP = λG + λH [25].
The λ2 -eigenfunctions of the Cartesian graph product uP are either uP = 1G ⊗ uH
with the second eigenvalue λ2,P = λ2,H or uP = uG ⊗ 1H with λ2,P = λ2,G , where
1G and 1H are the all-ones vectors on G and H, respectively. In case of degeneracy
λ2,P = λ2,H = λ2,G and both function products are λ2 -eigenfunctions of P = GH.
Lemma 2.5. Let P = GH be the Cartesian graph product of the graphs G and
H. If the λ2 -eigenspaces of G and H are tight, then the λ2 -eigenspace of P is also
tight.
Proof. Without loss of generality assume that λ2,G < λ2,H so that the λ2 eigenfunctions of P = GH are of the form uP = uG ⊗ 1H . The entry of uP at
the vertex (g, h) is therefore uP (g, h) = uG (g). Assume that uP is not tight, then
for some s one of the induced subgraphs P+ (uP , s) and P− (uP , s) is not connected
(the other one is connected by Lemma 2.3). Without loss of generality assume that
s > 0, so that the induced subgraph P+ (uP , s) is not connected. We now examine
G+ (uG , s), the subgraph induced by the same level s on G. From the construction of
P = GH and its λ2 -eigenfunction uP = uG ⊗ 1H follows that if a vertex of P belongs to the induced subgraph, (g, h) ∈ P+ (uP , s), then the corresponding vertex of G
belongs to the induced subgraph, g ∈ G+ (uG , s). Similarly, if (g, h) ∈ P− (uP , s) then
g ∈ G− (uG , s). An immediate consequence is that G+ (uG , s) is also not connected
since it can be connected only through vertices g such that belong to G− (uG , s),
contradicting our assumption that uG is tight.
The intuition underlying the proof is illustrated in Figure 2.3, where a λ2 eigenfunction uP = 1⊗uC20 of the Cartesian product of the cycles P = C10 C20 is plotted.
The projection of the eigenfunction on C20 is uC20 , the λ2 -eigenfunction of C20 . It is
evident that from the tightness of uC20 follows the tightness of uP .
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Since the Cartesian product is associative, Lemma 2.5 applies also for a product of any finite number of graphs P = G1 G2 . . . Gm . The λ2 -eigenspace of
P involves only the λ2 -eigenspaces of the graphs with the least second eigenvalue,
min{λ2 (G1 ), λ2 (G2 ) . . . λ2 (Gm )}, while the rest of the graphs contribute only a factor of 1. It is therefore clear from the latter proof that only those λ2 -eigenspaces need
to be tight to assure the tightness of the product’s λ2 -eigenspace. Another corollary
of Lemma 2.5 is:
Corollary 2.6. The λ2 -eigenspaces of a Cartesian product of paths, cycles and
complete graphs are tight.
Hamming graphs, for example, are products of complete graphs and their λ2 eigenfunctions are therefore tight.
We note that Lemmas 2.4 and 2.5 and their corollaries are only examples for
tight λ2 -eigenspaces. The link between tightness and the absence or presence of saddle points hints for more general tightness criteria. For example, it appears that
heterogeneity of the Laplacian weights ∆ij may induce critical points in the λ2 eigenfunctions and thus break tightness.
3. Tight polyhedral mappings and the multiplicity of the second eigenvalue. Next, we show that if the multiplicity of λ2 is m then the m-dimensional
λ2 -eigenspace spanned by the m eigenfunctions u1 , u2 . . . um defines a tight mapping
of G into the Euclidean space Rm . This therefore sets Banchoff’s upper bound on m.
Figure 3.1 shows the tight mapping of a triangulated torus of multiplicity m = 6.
Theorem 3.1. If the λ2 -eigenspace of a Laplacian on a graph G is tight then
its maximal multiplicity m̄(G) is bounded, m̄(G) ≤ chr(γ(G)) − 1, where γ(G) is the
genus of the graph and the chromatic number chr(γ) is given by Heawood’s formula,
(3.1)
1
7 + 49 − 24χ .
chr(γ) =
2
Proof. To construct the tight mapping we first embed the graph G into a
surface S of genus γ(G). The embedding is determined by enumerating the faces
of the surface. One method to specify the embedding, termed the rotation system, is to list the order of the edges around each of the vertices of G (for details
see [22]). The λ2 -eigenfunctions u1 , u2 . . . um ∈ RV define a mapping of S into
the m-dimensional Euclidean space, ϕ : S → Rm , by mapping each vertex v to
ϕ(v) = (u1 (v), u2 (v) . . . um (v)).
Case I: The mapping preserves edges, that is any two adjacent vertices α, β are
mapped to different points in Rm , ϕ(α) = ϕ(β). Each edge (α, β) is then mapped
to the vector stretched between the images of its two endpoint vertices, ϕ(α) and
ϕ(β), and the faces are mapped to planar polygons determined by their vertices. If
the vertices of a certain face are not coplanar then the face is triangulated. Since ϕ
preserves edges it is an immersion of S into Rm . Every direction in Rm corresponds
to a linear combination of the λ2 -eigenfunctions, which is also tight (Definition 2.1).
Thus, by the tightness of the λ2 -eigenspace the resulting immersion has the two-piece
property in any given direction and is therefore tight. By the linear independence
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of the ui , the image ϕ(S) is not contained in any hyperplane and the mapping is
therefore substantial. Banchoff proved that the maximal substantial dimension m of
the mapping is bounded m ≤ chr(γ(G))) − 1 [3, 24, 5], where the chromatic number
chr(γ(G)) is given by Heawood’s formula [29]. Since m is also the dimension of the
λ2 -eigenspace it proves the theorem for case I.
Case II: The mapping does not preserve edges, that is there exists at least one
edge (α, β) which is mapped to a point in Rm , ϕ(α) = ϕ(β). Thus ϕ fails to be
an immersion of G and S into Rm . To remedy this kind of pathology and make ϕ
an immersion (of a modified graph and surface) we apply the following contraction
procedure: We identify the vertices α and β by contracting the edge (α, β). The
genus of the resulting contracted graph Gc and the related contracted surface S c
either remains unchanged or decreases. We follow the contraction procedure until all
the pathologies are removed. The outcome is a polyhedral immersion, of Gc and S c
into Rm .
Fig. 3.1. The six λ2 -eigenfunctions u1 , u2 . . . u6 on a triangulated torus are represented
by grayscale (left). The non-diagonal entries of the Laplacian are ∆ij = ∆ji = −1 if i and j are
adjacent, otherwise ∆ij = 0, and the diagonal entries are all ∆ii = 6. The number of eigenfunctions
in this case is equal to the conjectured maximal multiplicity chr(1) − 1 = 6. The triangulated surface
(right) is a three-dimensional projection ϕp (v) = (u1 (v), u3 (v), u3 (v)) of the six-dimensional tight
mapping ϕ : S → R6 on three of the λ2 -eigenfunctions.
The ”contracted” functions uc1 , uc2 . . . ucm that correspond to the immersion are
obtained from the original λ2 -eigenfunctions ui by identifying their α and β entries
for every edge (α, β) that is mapped to a point. The linear independence of the uci
follows from that of the ui , since we only removed entries that were identical in all
the ui . The contracted functions are vectors of length equal to the number of vertices
V minus the number of pathological edges. This length is at least m by the linear
independence of uci .
To show that ϕ is indeed a tight immersion of Gc and S c , we need to show
that the uci define a tight function-space. The contracted functions uci span an mdimensional vector-space. Along each direction i in Rm the image uci (Gc ) is equal (as
a set) to the image ui (G). The tightness of uci therefore follows from the tightness
of ui . The same is true for any general direction (and linear combination of uci )
and the vector-space spanned by uci is therefore tight. By the linear independence
of the uci , the image ϕ(S c ) is not contained in any hyperplane and the mapping
is therefore substantial. The genus of the contracted graph and surface is equal
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or smaller than that of the original graph, γ(Gc ) ≤ γ(G). By Banchoff’s theorem
m ≤ chr(γ(Gc )) − 1 ≤ chr(γ(G)) − 1, where the last inequality follows from the
fact that the chromatic number in Heawood’s formula (Eq. 3.1) is a non-decreasing
function of the genus γ. Since m is the dimension of the λ2 -eigenspace it proves the
theorem for case II.
Theorem 3.1 sets Banchoff’s or Colin de Verdière’s bound on any tight eigenspace,
not necessarily that of λ2 . We also note that Theorem 3.1 is valid also for Schrödinger
operators, which are the discrete analogues of the general second-order self-adjoint
operators:
Corollary 3.2. Let H = ∆ + V be a Schrödinger operator on a graph G, where
V is the diagonal potential matrix. If an eigenspace of H is tight then its maximal
multiplicity m̄(G) is bounded, m̄(G) ≤ chr(γ(G)) − 1, where γ(G) is the genus of the
graph and the chromatic number chr(γ) is given by Heawood’s formula (Eq. 3.1).
This corollary follows from the applicability of Theorem 1.1 to Schrödinger operators [17, 21].
Similar to the definition of m̄(S), one may also define an upper bound m̃(γ) of the
λ2 -multiplicity m over all the graphs of a certain genus γ(G) whose λ2 -eigenspace is
tight. By Theorem 3.1 m̃(γ) ≤ chr(γ) − 1. For any γ one can construct the Laplacian
over the complete graph whose number of vertices is the chromatic number Kchr(γ).
The genus of this graph is γ(Kchr(γ)) = γ and its λ2 -eigenspace is tight (Corollary
2.2). The maximal m for this graph is m̄(Kchr(γ)) = chr(γ) − 1, which is achieved
when the weights of the Laplacian for all the edges are equal. From this follows
Corollary 3.3. m̃(γ), the upper bound on m over all the graphs of a certain
genus γ(G) whose λ2 -eigenspace are tight, is m̃(γ) = chr(γ) − 1.
In other words, Colin de Verdière’s conjecture [12] applies for this class of graphs.
Acknowledgment. I thank J. -P. Eckmann and Elisha Moses for many valuable
discussions and critical reading of the manuscript. The work was funded by the
Minerva foundation and the Center for Complexity Science.
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